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Put the car in diagnostic mode and run it. Watch for any codes (other than 51 which you'll set from pressing the throttle). It does sound quite a bit like limp mode. I can't think of anything pertaining to the transmission that would cause the car to respond like this, especially while stopped. It requires a significant problem for the ECU to enter limp mode... knock sensor fault, significant knock detected, sudden loss of oil pressure signal, faulty ECT, damaged wire harness and so on.

Fuel pressure is a possibility but I don't believe it is very likely... It won't hurt to check the pressure though.

Have you checked the mechanical timing and timing belt already?

I'll look back over this this weekend and see if/what I've missed and let you know if I can think of anything to check next. Worst case scenarios we start back at the top and work through the list.

No, Toyota does not make various thicknesses of exhaust gaskets. The gasket will compress somewhat after initial installation but nothing significant. The exhaust may just be hanging up on the studs in which case releasing the hangers may help.

How to make sure the hoses are tight enough.... hmm... No easy answer I'm afraid. If they're the right size you would just need to be sure the nipple is clean of buildup, corrosion and pitting and then use the proper constant tension clamp that Toyota put on at the factory. If you think the CT clamp has lost some of its elasticity then get another from Toyota. If you've switched to a worm gear hose clamp... well, just tighten until it stops. If it feels like it's too tight and is going to come apart, well, it probably is and there is something wrong with the nipple or the hose.

I know how you feel situations like this suck. You just need to make an actual list of how and what to check and in what order, set aside a few hours on Sat and Sun and hammer through it (TSRM in hand).